The following lemma is useful and follows quickly from Theorem 4.1.3. It shows that when you string these
finite variation curves together end to end, you could just as well save trouble on the details and consider a
single finite variation vector valued function.

Lemma 4.2.9In the above definition whereγk

(bk)

= γk+1

(ak)

, there exists a continuousbounded variation function,γone to one onγ−1

(γk [ak,bk))

which is defined on some closedinterval,

[c,d]

, such thatγ

([c,d])

= ∪k=1mγk

([ak,bk])

andγ

(c)

= γ1

(a1)

whileγ

(d)

= γm

(bm )

.Furthermore,

∫ m∑ ∫
f ⋅dγ = f ⋅dγk. (4.19)
γ k=1 γk

(4.19)

Ifγ :

[a,b]

→ ℝpis of bounded variation and continuous, then

∫ ∫
f ⋅dγ = − f ⋅dγ. (4.20)
γ − γ

(4.20)

Proof: Consider the first claim about the intervals. It is obvious if m = 1. Suppose then
that it holds for m − 1 and you have m intervals and curves. By induction, there exists